#243899

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,5,7,4,8)(9,10,13,19,14,18,15,29)(11,20,23,32,22,36,24,16)(12,25,27,35,28,17,30,34)(21,37,26,40,38,33,39,31);(1,2)(3,8)(4,6)(5,7)(9,11,15,23,14,22,13,24)(10,16,29,20,18,32,19,36)(12,26,30,38,28,39,27,21)(17,33,35,31,25,37,34,40);(2,6)(3,4)(7,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,30)(16,31)(18,25)(19,34)(20,37)(22,38)(23,26)(24,39)(29,35)(32,40)(33,36);(2,7)(6,8)(10,18)(11,22)(17,25)(19,29)(21,38)(23,24)(26,39)(34,35)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151095);